Finite Group Databasebeta

← #27153605

Subgroups — #27153605

Order 12416.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261443C211yesmaximal, characteristic
2Q8×C22⋊D97C211yesmaximal, characteristic
3#20261084C211yesmaximal, characteristic, complemented
4#20261071C211yesmaximal, characteristic, complemented
5#20261069C211yesmaximal, characteristic, complemented
6#20261028C211yesmaximal, characteristic, complemented
7#20260974C211yesmaximal, characteristic, complemented
8#20260973C211yesmaximal, characteristic, complemented
9#20260943C211yesmaximal, characteristic, complemented
10#20260889C211yesmaximal, characteristic, complemented
11Q16×Dic97C211yesmaximal, characteristic, complemented
12#20260798C211yesmaximal, characteristic
13C2×#15690734C211yesmaximal, characteristic, complemented
14#103×D97C211yesmaximal, characteristic, complemented
15#451×C97C211yesmaximal, characteristic, complemented
16C2×Q8×D97C2211yescharacteristic
17C2×#13095501C2211yescharacteristic
18C22×#11715895C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic
20#15690809C2211yescharacteristic
21#15690808—21no—
22#15690808C2211yescharacteristic
23#15690800—21no—
24#15690790—21no—
25#15690789—21no—
26#15690789C2211yescharacteristic
27#15690787—21no—
28#15690786—21no—
29#15690786C2211yescharacteristic
30#15690786C2211yescharacteristic
31Q8×Dic97C2211yescharacteristic
32Q8×Dic97C2211yescharacteristic
33C4×C22⋊D97—21no—
34C4×C22⋊D97C2211yescharacteristic
35#15690770—21no—
36C2×C4⋊Dic97C2211yescharacteristic
37C4⋊C4×D97C2211yescharacteristic
38#15690744C2211yescharacteristic
39#15690744C2211yescharacteristic
40#15690743C2211yescharacteristic
41#15690739C2211yescharacteristic
42#15690735C2211yescharacteristic
43#15690735C2211yescharacteristic
44#15690734C2211yescharacteristic
45#15690734C2211yescharacteristic
46#15690734C2211yescharacteristic
47#15690734C2211yescharacteristic
48#15690734C2211yescharacteristic
49#15690733C2211yescharacteristic
50#15690722C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C22×Q8×C97C2211yescharacteristic
53C2×Q16×C97C2211yescharacteristic
54C2×Q16×C97C2211yescharacteristic
55C22⋊Q8×C97C2211yescharacteristic
56#103×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58C22⋊C8×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194—21no—
61C2×Dic194C2311yescharacteristic
62Q8×D97—22no—
63Q8×D97—41no—
64C2×C22⋊D97D411yescharacteristic
65C22×Dic97—21no—
66C2×C4×D97—21no—
67C2×C4×D97—21no—
68C2×C4×D97C2311yescharacteristic
69#13095501—41no—
70C2×#11715895—41no—
71C2×#11715895C2311yescharacteristic
72C2×#11715895C2311yescharacteristic
73C8×D97—41no—
74C4⋊Dic97—21no—
75C4⋊Dic97—21no—
76C4⋊Dic97—21no—
77C4⋊Dic97C2311yescharacteristic
78C4⋊Dic97C2311yescharacteristic
79C4⋊Dic97C2311yescharacteristic
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83#13095490—21no—
84#13095490—21no—
85#13095490—21no—
86#13095490—21no—
87#13095490D411yescharacteristic
88C22⋊Dic97—21no—
89C22⋊Dic97D411yescharacteristic
90#13095488—21no—
91#13095488—21no—
92#13095488D411yescharacteristic
93C4×Dic97—21no—
94C4×Dic97—21no—
95C4×Dic97C2311yescharacteristic
96C2×Q8×C97—21no—
97C2×Q8×C97—22no—
98C2×Q8×C97—41no—
99C2×Q8×C97C2311yescharacteristic
100C2×Q8×C97C2311yescharacteristic
101C2×Q8×C97C2311yescharacteristic
102C22×C4×C97—21no—
103C22×C4×C97C2311yescharacteristic
104Q16×C97—22no—
105Q16×C97—41no—
106C2×C8×C97C2311yescharacteristic
107C2×C8×C97C2311yescharacteristic
108C4⋊C4×C97—21no—
109C4⋊C4×C97C2311yescharacteristic
110C22⋊C4×C97—21no—
111C23×C97C2×D411yescharacteristic
112C22×D97C2×D411yescharacteristic
113Q8×C97—21no—
114Q8×C97—21no—
115Q8×C97—21no—
116Q8×C97—21no—
117Q8×C97—21no—
118Q8×C97—42no—
119Q8×C97#4112yes—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—42no—
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2411yescharacteristic
129C22⋊D97Q1614yescomplemented
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—81no—
135C2×Dic97C2×D411yescharacteristic
136C2×Dic97C2×D411yescharacteristic
137C2×Dic97C2×D411yescharacteristic
138C4×D97—21no—
139C4×D97—21no—
140C4×D97—42no—
141C4×D97—81no—
142Dic194—21no—
143Dic194—21no—
144Dic194—42no—
145Dic194—81no—
146C8×C97—21no—
147C8×C97—21no—
148#11715895—21no—
149#11715895—21no—
150C22×C97—21no—
151C22×C97—21no—
152C22×C97C2×Q1612yes—
153C22×C97C22×D411yescharacteristic
154C2×D97—21no—
155C2×D97—21no—
156C2×D97C2×Q1612yes—
157C4×C97—22no—
158C4×C97—21no—
159C4×C97—41no—
160C4×C97—41no—
161C4×C97—41no—
162C4×C97C2×#4111yescharacteristic
163C4×C97H(5,𝔽2)11yescharacteristic
164Dic97—21no—
165Dic97—21no—
166Dic97—41no—
167Dic97—41no—
168Dic97—41no—
169Dic97—41no—
170Dic97C2×Q1612yes—
171C2×C97—22no—
172C2×C97#54611yescharacteristic
173C2×C97C22×Q1611yescharacteristic
174C2×C97#57711yescharacteristic
175D97—22no—
176#3299—971nomaximal, Sylow
177C97#329911yescharacteristic, complemented, Sylow
178#554—971no—
179D4×Q8—971no—
180#487—971no—
181#484—971no—
182#484—971no—
183#477—971no—
184#467—971no—
185#467—971no—
186#462—971no—
187#451—971no—
188#451D9711yescharacteristic, complemented
189C4×Q16—971no—
190C8×D4—971no—
191C2×#103—971no—
192C2×#103—971no—
193C22×Q8—971no—
194C22×Q8C2×D9711yescharacteristic
195C2×Q16—971no—
196C2×Q16C2×D9711yescharacteristic
197C2×Q16C2×D9711yescharacteristic
198C22×C8—971no—
199C22×C8—971no—
200C4⋊Q8—971no—
201C4⋊Q8—971no—
202C4⋊Q8—1941no—
203#125—1941no—
204C22⋊Q8—971no—
205C22⋊Q8—971no—
206C22⋊Q8—971no—
207C22⋊Q8—1941no—
208C22⋊Q8—1941no—
209C22⋊Q8—1941no—
210C22⋊Q8—1941no—
211C22⋊Q8C2×D9711yescharacteristic
212C4×Q8—971no—
213C4×Q8—971no—
214C4×D4—971no—
215C4×D4—1941no—
216C4×D4—1941no—
217C2×C4⋊C4—971no—
218C2×C4⋊C4—971no—
219#107—971no—
220#107—971no—
221#107—971no—
222C4⋊C8—971no—
223#103—971no—
224#103—971no—
225#103—971no—
226#103—971no—
227#103—971no—
228#103—971no—
229#103—971no—
230#103—971no—
231#103C2×D9711yescharacteristic
232#103C2×D9711yescharacteristic
233C22⋊C8—971no—
234C22⋊C8C2×D9711yescharacteristic
235C4×C8—971no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—971no—
240C2×Q8—3881no—
241C2×Q8—1941no—
242C2×Q8—1941no—
243C2×Q8—1942no—
244C2×Q8C22×D9711yescharacteristic
245C2×Q8C22×D9711yescharacteristic
246C2×Q8C22×D9711yescharacteristic
247C2×D4—971no—
248C22×C4—21no—
249C22×C4—971no—
250C22×C4—1941no—
251C22×C4—1941no—
252C22×C4—1941no—
253C22×C4C22×D9711yescharacteristic
254Q16—22no—
255Q16—41no—
256Q16—3881no—
257C2×C8—971no—
258C2×C8—971no—
259C2×C8—3881no—
260C2×C8—3881no—
261C2×C8C22×D9711yescharacteristic
262C2×C8C22×D9711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—971no—
265C4⋊C4—971no—
266C4⋊C4—971no—
267C4⋊C4—971no—
268C4⋊C4—1941no—
269C4⋊C4—1941no—
270C4⋊C4—1941no—
271C4⋊C4—1941no—
272C4⋊C4—1941no—
273C4⋊C4—1941no—
274C4⋊C4—1941no—
275C4⋊C4—1941no—
276C4⋊C4—1941no—
277C4⋊C4—1941no—
278C4⋊C4C22×D9711yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—971no—
281C22⋊C4—971no—
282C22⋊C4—1941no—
283C22⋊C4—1941no—
284C22⋊C4—1941no—
285C42—971no—
286C42—1941no—
287C42—1941no—
288C23—971no—
289C23D4×D9711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—7761no—
297Q8—3882no—
298Q8—1941no—
299Q8—1941no—
300Q8#1309551012yes—
301D4—974no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—7761no—
310C2×C4—7761no—
311C2×C4—971no—
312C2×C4—971no—
313C2×C4—971no—
314C2×C4—3882no—
315C2×C4—1941no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4—1941no—
321C2×C4D4×D9711yescharacteristic
322C2×C4C23×D9711yescharacteristic
323C8—21no—
324C8—21no—
325C8—1941no—
326C8—1941no—
327C22—21no—
328C22—21no—
329C22—972no—
330C22—1941no—
331C22—1941no—
332C22Q16×D9712yes—
333C22C2×D4×D9711yescharacteristic
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—972no—
340C4—3881no—
341C4—3881no—
342C4—3881no—
343C4—3881no—
344C4—1941no—
345C4—1941no—
346C4C2×#1309551011yescharacteristic
347C4#1569085511yescharacteristic
348C2—22no—
349C2—1942no—
350C2#2026137211yescharacteristic
351C2C2×Q16×D9711yescharacteristic
352C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#451×C97C2
socleC22×C97C22×D4
prime radicalC2×C22⋊D97D4
hypercenter#451D97
hypocenterC97#3299
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20261084C2semidirect
4. #20261071C2semidirect
5. #20261069C2semidirect
6. #20261028C2semidirect
7. #20260974C2semidirect
8. #20260973C2semidirect
9. #20260943C2semidirect
10. #20260889C2semidirect
11. Q16×Dic97C2semidirect
13. C2×#15690734C2semidirect
14. #103×D97C2semidirect
15. #451×C97C2semidirect
129. C22⋊D97Q16semidirect
177. C97#3299semidirect
188. #451D97semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202614432
Q8×C22⋊D972
#202610842
#202610712
#202610692
#202610282
#202609742
#202609732
#202609432
#202608892
Q16×Dic972
#202607982
C2×#156907342
#103×D972
#451×C972
#329997